Let F : R 2 R be continuous, where R 2 is equipped with the Euclidean metric d2. Let a, b R be so that a

Exercise 6 (10 points). Let F: R2 > R be continuous, where R2 is equipped with the Euclidean metric d2. Let a,b 6 R be so that a. '6'([a, b]; R) the function dened by K(f)(:c) = / F(t, f(t))dt for 3.11 f e %([a,b];R) and a: 6 [a,b]. Show that there exists a unique 9 E '6( [(1, b]; R) such that g'(:c) = F(:c, 9(a)) for all a: E ((1,5) and 9(a) = 0, that is, Show that the differential equation cp' = F(x, (,0) with initial condition 90(a.) = 0 has a unique solution on ((1,6). [Hint Look back at Exercise 4 in Homework 2.]